Sunday, June 14, 2015

In 2008, the Gallop organization randomly called around 355,000
people in the United States, asking them about their state of happiness.They wanted to quantify how this state
changed as people transitioned from youth, to middle age, to old age.To assess global well-being, they asked the
following:

“Please imagine a ladder with steps numbered from 0 at the
bottom to 10 at the top. The top of the
ladder represents the best possible life for you, and the bottom of the ladder
represents the worst possible life for you.
On which step of the ladder would you say you personally feel you stand
at this time?”

They found that youngest people, those in their late teens
and early 20s, felt quite upbeat about their life, placing themselves high up
on the ladder. Unfortunately, this sense
of well-being dropped as age increased, reaching its lowest point around the age
of 50. But as age increased beyond 50,
the sense of well-being increased dramatically, continuing to grow even into the 8th and 9th decade of life.

Well-being ladder as a function of age in America, as assessed in 2008

The trend was consistent in both men and women.And so amazingly, people in America felt that
they reached their lowest point on the ladder of life around the time they
reached midlife.What is it about aging
beyond the teen years that made people feel worse about their sense of well-being, and why did this process reverse in the fifth decade of life?

Stress and worry
decline in mid life

Statistical analysis of the data revealed that some obvious
things that one might think affects sense of well-being did not alter the
age-dependent process. For example,
gender, being unemployed, having a child living at home, and not having a partner
had no significant effects on the trends.
That is, regardless of these factors, people simply became less happy as
they aged toward midlife, and then something seemed to change, allowing them to
regain happiness as they got older.

The authors of the study, writing in the Proceedings of the National Academy of
Science, speculated that perhaps this change had something to do with
increased wisdom and emotional intelligence of the aged. They wrote: “older people have an increased
ability to self-regulate their emotions and view their situations positively.”

They provided some data to back their speculation. That same Gallop poll had also asked the respondents
to evaluate how they felt yesterday. They asked about
specific affects like worry, stress, and anger: “Did you experience stress
during a lot of the day yesterday? Did you
experience worry during a lot of the day yesterday?”

They found that regardless of age, women reported that their yesterday had greater stress and worry than men. However, there was a dramatic age-dependent effect. When young people evaluated their yesterday, they reported much more stress and worry than older people. But for people in their late 40s and early 50s, their yesterday was suddenly much less stressful than for people 10 years younger. For people in their 60s, their yesterday was even more peaceful.

The emotional
response to what could've been

This questions of why people (at least Americans) feel a reduced
sense of wellbeing with age, and why this process reverses at midlife, remain unanswered. One intriguing line of research is that with
age, the brain alters how it evaluates lost opportunities.

Stefanie Brassen and colleagues asked a group of young
people (around 25 years old), a group of healthy older people (around 66 years
old), and a group of late-life depressed elderly (also around 66 years old), to
watch a monitor which showed 8 boxes.
Seven of those boxes contained gold, but one had a devil in it. Boxes could be opened in sequence, and as
long as the box contained gold, they could keep accumulating the reward. But if the box contained a devil, they would lose
everything. So the participants could
decide to stop and collect their gains, or continue. Importantly, if they decided to stop, the
position of the devil was revealed. This
indicated how far they could have safely continued, thereby showing them the
missed opportunity.

Young people responded to the missed opportunity by being aggressive
on the next trial—the greater the missed opportunity on the current trial, the
greater the risk that they took on their next attempt. Surprisingly, following a missed opportunity
the depressed elderly did the same as the young people, taking a bigger risk. However, healthy elderly did not respond this
way. Following a missed opportunity,
they did not increase their risk taking.

To measure the emotional response to the missed opportunities,
the authors measured skin conductance and found that this measure was modulated
in the depressed elderly but not in the healthy elderly. While these data were being collected, the
authors also measured brain activity using fMRI and found that whereas all
groups responded similarly to winning and losing, the main difference was in
the response to a missed opportunity. Both
the young and the depressed elderly had a strong response when they observed
the missed opportunity. The healthy
elderly, however, only responded to real losses, and not missed opportunities.

The data suggested that healthy aging was associated with a
reduced responsiveness to lost opportunities.
When the healthy elderly made their decision, they were happy if they
gained something, and did not care so much if that gain was less than
optimal. That is, they did not respond emotionally
to the fact that they could have made even a better decision.

On the other hand, both the young folks and the depressed
elderly responded emotionally once they found out that they could have made a
better decision, despite the fact that the decision that they had made had
produced a gain. That gain, in
retrospect, was not good enough. And in
some ways not being as good as it could have been felt like a loss to the
young folks and the depressed elderly.

So it is possible that in youth, a decision that results in a positive outcome, but represents a lost opportunity (because it could have been better), produces an emotional, stressful response. But when that same decision is made after midlife, the brain is less sensitive to the fact that the results could have been better, and more concerned with the fact that the decision produced a gain. Curiously, this is true for elderly who are healthy, but not elderly who are depressed.

In his poem, "The Bridge", Henry Wadsworth Longfellow writes:

For my heart was hot and restless, And my life was full of care,And the burden laid upon me Seemed greater than I could bear.

But now it has fallen from me, It is buried in the sea;And only the sorrow of others Throws its shadow over me.

Yet whenever I cross the river On its bridge with wooden piers,Like the odor of brine from the ocean Comes the thought of other years.

And I think how many thousands Of care-encumbered men,Each bearing his burden of sorrow, Have crossed the bridge since then.

I see the long procession Still passing to and fro,The young heart hot and restless, And the old subdued and slow!

And forever and forever, As long as the river flows,As long as the heart has passions, As long as life has woes;

The moon and its broken reflection And its shadows shall appear,As the symbol of love in heaven, And its wavering image here.

Stone, A. A., Schwartz, J. E., Broderick, J. E., & Deaton,
A. (2010). A snapshot of the age distribution of psychological well-being in
the United States.Proceedings of the National
Academy of Sciences,107:9985-9990.

Sunday, June 7, 2015

The ancient Greeks, along with a number of other
civilizations, noticed five “wandering stars” that over many nights appeared to
travel against the background of fixed stars in the sky. These “stars” went along the same path as the
Sun and the moon, but in the opposite direction. They named the wandering stars Hermes,
Aphrodite, Ares, Zeus, and Cronos. The
Romans translated these names into Mercury, Venus, Mars, Jupiter, and
Saturn. The names of the wandering
stars, along with the Sun and the moon, became the names of the 7 days of the
week. Saturday, Sunday, and Monday are
associated with Saturn, the Sun, and the moon.
Tuesday is thought to be associated with the Germanic god Tyr (Mars),
Wednesday with Wotan (Mercury), Thursday with Thor (Jupiter), and Friday with
Frigga (Venus).

The wandering stars, of course, were no stars at all, but
planets. It took about two thousand
years to understand why the planets appeared to wander. The story begins around the time of Aristotle,
and ends with Newton. Along the way, humans
learned how to use mathematics to represent observations in nature, and this
led to the birth of science. In a recent
book titled “To Explain the World”, Stephen Weinberg, a physicist and Noble
Laureate, tells this story. Here, I
simplify his eloquent and thorough text, and highlight the key ideas.

Aristotle and Ptolemy

Anaxagoras, an Ionian Greek born around 500 BC, reasoned
that the earth is spherical because when the Sun placed the earth’s shadow on
the moon, one could see the round outline of the earth. Aristotle repeated this idea in his book “On
the Heavens”, writing: “In eclipses the outline is always curved, and, since it
is the interposition of the Earth that makes the eclipse, the form of the line
will be caused by the form of the Earth’s surface, which is therefore
spherical.” But he also argued that the
earth must be stationary and not moving, because if it were moving a rock
thrown upward would not fall straight down, but to one side. He wrote: “heavy bodies forcibly thrown quite
straight upward return to the point from which they started, even if they are
thrown to an unlimited distance.”

Given that the earth is not moving, how does one explain the
fixed and the wandering stars (the planets)?
Aristotle, citing an earlier work by Eudoxus of Cnidus, suggested that
the fixed stars are carried around the earth on a sphere that revolves once a
day from east to west, while the sun and moon and planets are carried around
the earth on separate (and transparent) spheres. Now there were lots of problems with this
scheme. For example, because the planets
were thought to shine with their own light, and the spheres were always the
same distance from the earth, the brightness of the planets should not change,
which disagreed with observations.

This issue remained unresolved until 650 years later, with
Claudius Ptolemy, who in AD 150, working in Alexandria, Egypt, wrote Almagest. Ptolemy gave up on the notion that earth was
the center of rotation for the planets, and instead suggested that each planet
had a center of rotation that itself went around the earth. For the nearby planets of Venus and Mercury,
he proposed that the centers of rotation were always along a line between the
earth and the sun, and went around the earth in exactly one year. For Mars, Jupiter, and Saturn, the centers of
rotation were beyond the sun.

Ptolemy's planetary model

Ptolemy wrote: “I know that I am mortal and the creature of a day; but when I search out the massed wheeling circles of the stars, my feet no longer touch the earth, but, side by side with Zeus himself, I take my fill of ambrosia, the food of the gods.”

Copernicus and Tycho Brahe

For centuries the idea that the earth was stationary
remained, so that even in the middle ages, scholars like Jean Buridan would
reject the idea that the earth could be rotating, not realizing that if earth
rotated, then its rotation would give everything, including an arrow that was
shot straight up, an impetus. Like all
good mentors, Buridan had a student who thought independently. His name was Nicole Oresme. Oresme studied with his mentor Buridan in
Paris in 1340s. In his book “On the
Heavens and the Earth”, Oresme rejected Aristotle’s arguments for a stationary
earth, stating that when an archer shoots an arrow vertically, the earth’s
rotation carries the arrow with it (along with the archer). Therefore this observation is not a
demonstration of an immovable earth, but also consistent with a rotating
earth. Aristotle’s argument on a
stationary earth took its first major blow.

The idea that the earth might be rotating took center stage
with Nicolaus Copernicus, who in 1510 wrote a short, anonymous book titled
“Little Commentary”. The book was not
published until after the author’s death, but in it he put forth a new
theory. He began by asserting that there
is no center for the orbits of the celestial bodies: the moon goes around the
earth, but all other heavenly bodies go around a point near the sun. He further asserted that the night sky has
fixed stars that are much farther away than the sun, and appear to move around
the earth only because the earth is rotating on its axis and about the sun.

Tycho Brahe was impressed with the simplicity of Copernicus’
theory, but pointed out a huge problem:
if the earth is moving, what is moving it? After all, earth was made of rocks and dirt, materials
that would make something the size of earth weigh an enormous amount. In contrast, ever since Aristotle it was
thought that the heavenly bodies were nothing like earth, made of some kind of
substance that gave them a natural tendency to undergo rapid circular
motion. The problem was, if earth was
moving around the sun, what was pushing it, and what was keeping it there in
its orbit?

In an ironic twist, to explain motion of the earth it was
the Copernican astronomers who called on divine intervention. In a letter to Brahe, Copernican Christoph
Rothmann wrote: “These things that vulgar sorts see as absurd at first glance
are not easily charged with absurdity, for in fact divine Sapience and Majesty
are far greater than they understand.”

Being unimpressed with divine intervention, in 1588 Tycho
Brahe pointed out that if one took Ptolemy’s theory and put the moving center of
all the planets (except earth) on the sun, and have the sun go around the
stationary earth, then much of the observed data would fit just as well as
Copernicus’ theory. This “Tychonic”
system kept the advantage of a stationary earth, and was mathematically
identical to the model of Copernicus.

Tycho Brahe's planetary model

In January of 1610, Galileo used his newly built telescope
to look at Jupiter, and saw that “three little stars were positioned near him,
small but very bright.”The next night
he noticed that the little stars seemed to have moved, and eventually he
concluded that the little stars were actually satellites of Jupiter, its
moons.This observation was critical, as
it was the first discovery of heavenly objects that circled something other
than earth.They were a miniature
example of what Copernicus had proposed.But Tycho Brahe’s model remained a viable alternative, because the fundamental
question for a sun-centric theory remained that if the earth is moving, what could
be so powerful as to move it?

Newton and calculus

In 1665, Issac Newton asked a simple question: how
does one compute speed of some object if the distance traveled as a function of
time is not constant (or uniform).
Suppose x(t) represents position
as a function time t.Newton argued that
in order to calculate speed, we need to think of an infinitesimally small
period of time, which he called o. Speed becomes:

For example, suppose that

For
o an infinitesimal period of time, we
can ignore terms that include squared and cubic powers of o.This means
that:

Newton called this the "fluxion" of x(t). We now call it the derivative of x(t).

Newton was considering this question because he wanted
to ask about the acceleration that a body would experience as it travels in
constant speed about a circle. At any
time t, the velocity of this body is a vector tangent to the circle, with
amplitude v.

Suppose that the circle is
radius r. After an infinitesimal time o, the body will
have traveled by a distance vo, and
angle q about the circle. At this new location the speed would still be
v, but the velocity vector will have rotated by an angle q. We
now have two isosceles triangles that are scaled versions of each other. Therefore, the ratio of the short side to the
long side of the two triangles is equal:

We can re-write the above equation as follows:

Eq. (1)

The
term on the left of the above equation is a derivative. It represents the length of the acceleration
vector that the body experiences (pointing to the center of the circle) as it
rotates with constant speed around the circle.

Newton realized that this acceleration toward the center is due to a force that is pulling
the body toward the center of the circle (otherwise, it would fly off in a
straight line, tangent to the circle). That force, he assumed, is proportional to square of the velocity v, divided by radius r.

Next,
Newton considered Kepler’s observation (his third law) that the square of the
period of a planet in its orbits is proportional to the cube of the radius of
its orbit. The period of a body moving
with speed v around a circle of radius r is the circumference 2pr divided by speed v. And so
Kepler’s third law says that

We
can re-write the above equation as follows:

Eq. (2)

If we now compare Eq. (1) with Eq. (2), we see that the acceleration that was keeping the body moving in circular motion, is also proportional to the reciprocal of squared r. This means that the force that is pulling the body toward the center is proportional to the inverse of the squared distance of the body from the center. This is the inverse square law of gravity.

But
the incredible discovery was still one step away. Newton now asked whether the acceleration of
the moon in its orbit around the earth is the same acceleration that a body
undergoes when it is falling here on earth. To calculate moon’s acceleration, he estimated the distance of the moon
to the center of the earth to be around 60 times the radius of earth, or around
314 million meters. Next, he computed
the speed of the moon by dividing the circumference of one orbit around the
earth by its period of travel (27.3 days, or 2.36 million seconds):

He then used Eq. (1) to compute the acceleration of the
moon toward the earth:

This is the moon’s acceleration toward earth. It is quite small, but Newton understood that
the acceleration is small because the moon is very far away from earth. An object on the surface of earth accelerates
faster because it is at a distance of one radius of earth away from the earth’s
center. The moon is 60 times
farther. Therefore, using Eq. (2), he
argued that the moon’s acceleration should be 1/60^2 that of an object on the
surface of the earth.

Multiplying moon’s
acceleration by 60^2 we find the result that a body on the surface of
earth should accelerate at around 8 m/s^2.
(The actual value is 9.8 m/s^2. The
greatest source of error in Newton’s calculations was that the distance of moon
from earth, which he underestimated by around 15%). He then writes:

“I began to think of gravity extending to the orb of the
moon and (having found out how to estimate the force with which [a] globe
revolving within a sphere presses the surface of the sphere) from Kepler’s rule
of the periodical times of the plants being in sesquilterate proportion of
their distances from the center of the orbs, I deduced that the forces which
keep the planets in their orbs must [be] reciprocally as the squares of their
distances from the centers about which they revolved and thereby compared the
moon in her orb with the force of gravity at the surface of the earth and found
them answer pretty nearly.”

So what Newton had done was to show that the motion of the
moon around the earth described an acceleration toward earth that was due to a
force quite identical to the force that acts on an apple on the surface of the
earth. The only reason that the moon accelerates
much slower toward earth is because the moon is much farther, and therefore the
force that it feels from earth is much weaker.

The acceleration of the apple, the moon, and the planets around the sun,
are all governed by the same rules: force grows weaker as the squared distance
of one body from another.

Follow by Email

About Me

I was born in Iran and immigrated to the US at the age of 14. I was educated at Gonzaga University, University of Southern California, and finally MIT. I studied under the mentorship of Prof. Michael Arbib and Prof. Emilio Bizzi. I am currently Professor of Biomedical Engineering and Neuroscience, and the Director of the BME PhD Program at Johns Hopkins School of Medicine. I am a neuroscientist who uses mathematics to understand how the brain controls our movements.